pragmatic inferences about actions - douglas walton's

DOUGLAS N. WALTON

PRAGMATIC INFERENCES ABOUT ACTIONS *

Does it matter whether natural conversations about human actions andconsequences of actions ever conform to formal principles of logicalinference? After all, was Hume not justified in his skeptical positionthat contingent matters of fact do not admit of necessary connections?What more needs to be said about the extent to which actions may besaid to be logical?

One problem stems from a long-standing member of the informaltraditional fallacies, the so-called circumstantial ad hominem fallacy.Suppose Dr. Smith delivers a small lecture to patient Jones on theadverse effects of smoking, citing statistical and other results to proveto Jones that several dangerous diseases. such as cancer of the lungsand throat, are directly caused by smoking. Dr. Smith concludes thatone should not smoke, and makes it clear that what she means to say isthat Jones should stop smoking. Suppose additionally that during thecourse of this conversation Dr. Smith is herself smoking a cigarette,blowing out plumes of smoke to punctuate her points. The tradition isthat Dr. Smith's performance may be open to an allegation of circum-stantial ad horninem by Jones' reply, "How can you say that one shouldnot smoke while you yourself smoke? Isn't that inconsistent?" Theproblem: Is Jones' criticism a reasonable one?

This problem is compounded by the initial difficulty that we need tosort out how much of what Smith has put forward is to count as Smith'sargument. For example, perhaps the medical and statistical evidencecited by Smith for the conclusion that smoking is harmful to health isgood evidence. If this much of the argument is acceptable to Jones,then Jones is certainly incorrect (and perhaps commits a form of ad hominem or ad feminan) to reject it simply because the immediatesource of it herself smokes.

But is that all there is to the argument? Surely the ultimate conclusionof Dr. Smith's argument, 'One should not smoke,' is a normative ratherthan purely statistical or medical conclusion. This normative conclusioncontains a directive to a certain course of action. It is not therefore fairto count as part of the argument Dr. Smith's own personal advocacy of

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that conclusion, in particular her own performance of smoking whilepropounding the argument that one should not smoke? The questionturns on whether the performance of an action can count as part of theargument. We seem to be at the pragmatic edge, if not right in thepragmatic wastebasket.

Here, then, is the disputed question. Some critics would say that thereis no logical inconsistency in condemning smoking while one smokes.Hence they would say that the ad hominem allegation is trivial orpointless. Critics would recognize the legitimacy of a deontic-action-theoretic sort of inconsistency in the conjunction of "Everybody oughtnot to smoke" with the present act of the speaker's smoking. To theextent to which one's actions may conflict with one's directives tocertain types of action, one's position as advocate of an argumentshould be open to serious ad hominem criticism, according to thisviewpoint.

Which side of this disputed question is more defensible depends onthe extent to which actions may be said to be logical. So it does mattervery much for our understanding of how arguments are criticizedwhether conversations about actions conform to principles of logicalinference.

A primary objective of our work will be to study conditionals of theform 'If an agent brings something about then she also brings some-thing else about'. Such conditionals are important to study for manyreasons, but we will particularly look to see how they function in whatGoldman (1970) has called level-generation of actions. We want toconfirm Goldman's thesis that more than one kind of conditional isinvolved, and show that these conditionals have similar, but by nomeans identical, properties. The logical structures appropriate to thetwo conditionals primarily studied turn out to be applications, respec-tively, of the relatedness logics and dependence logics of Epstein (1979,1980). That is not too surprising in the first case, because relatednesslogic was originally constructed to model conditionals in act-sequences,as shown in Walton (1979). However, the reader will see that somesignificant details of this original modelling have been changed in thatpart of the analysis given below that pertains to relatedness.

1. ANALYSIS OF ACT-SEQUENCES

According to Goldman (1970, ch. 2) the relationship that obtainsbetween pairs of act-tokens, and that allows act-sequences to be linked

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together into act-trees, is called level-generation. . Four categories oflevel-generation are distinguished: (1) causal generation, (2) con-ventional generation, (3) simple generation, (4) augmentation genera-tion. Conventional generation is characterized by the existence of rules,conventions, or social practices that link one action to another in aconditional, e.g., If John extends his arm out the car window then Johnsignals a turn. In simple generation, circumstances combine with oneaction to ensure the performance of another action, e.g., If Johndangles a line in the water then John is fishing.

Our concern here will not be directly on rules or backgroundcircumstances.

We will concentrate on the basic categories (1) and (4)

as types of conditionals in act-sequences, and leave the application ofour analysis of (2) and (3) to the reader. We will call our version of (1)external generation, and our version of (4) internal generation ofactions.

External generation of actions has often been called the accordioneffect. Consider this familiar sequence; Smith's moving his finger,Smith's flipping the switch, Smith's turning on the light, Smith'swarning a prowler. Each step in the sequence is an action, and somewould say the whole sequence is also an action. As has often beenremarked on, we can begin with a sequence of natural states of theworld - the finger movement, the switch-flipping, the light going on, theprowler's being warned - and the element of human agency is trans-mitted over that sequence by its initial appearance at the first step. Likethe other kinds of act-sequences studied by Goldman, this one fallsnaturally into an order as given above. But this sequence is ampliativeand extrinsic-directed, stretched out by nature into different space-timeregions, unlike the internalized and intrinsic-directed sequence ofprogressive containments of (4). The rationale of an external sequencewould seem to be that as each step in the description of the complex ofevents is taken, a new element yet one that is still somehow related tothe previous step is introduced. That is, each event is related to anevent that is immediately proximate to it in the sequence in such afashion that the separate links fuse together to make up a chain-likesequence. Each member stands in such a relationship of having to dowith the activation of its immediate neighbours that an orderlysequence of action is possible over the complex of individual actionsthus ordered. How each stage of the sequence is made to happen hassomething to do with how each other stage is made to happen. But whatdoes "has something to do with" mean in this context?

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Before we can answer this question, we have to ask what is related bythis relation in an act-sequence, i.e., what is an action? The firstpresumption of the present theory is that an action-sentence can berestated (clarified, analysed) in such a way that there is a propositionexpressed that is said to be made true by the agent. For example, "Sueputs the pencil on the desk" must be analysable by some form ofparaphrase like the following: "The pencil is on the desk", a pro-position made true by Sue. This form of analysis was originally due toSt. Anselm - see Henry (1967, pp. 120ff.) 1 - and has recently beendeveloped by (1977) and Walton (1979).

As Davidson (1967) pointed out, it is a nontrivial task to see howcommonplace action sentences could be made to conform to such aparaphrase. For example, "Cass walked to the station" cannot beprecisely equated with either "Cass made it true that she is at thestation" or "Cass made it true that she walked to the station". One wayof confronting this problem is to introduce a class of propositions called"action propositions" by Walton (1980) or "act-relations" by(1977) which meet this requirement: necessarily (p if, and only if, p ismade true), and p is possibly true.2 Since we may presume that it ispossible that Cass has two legs even though she has in no way made thistrue, "Cass has two legs" is not an action proposition. Whereas if Cassmade it true that she walked to the station, then "Cass walked to thestation" is an action proposition. So in general we presume such aparaphrase of action propositions is a feasible project. That part of theproject having to do with propositional calculus will comprise thesubject of this paper.

The expressive capacity of the language of actions we construct islimited to the bringing about of changes or leaving the status quo in agiven situation. For example, if the door is in fact closed, my options asagent are limited to (a) bringing it about that it is open, or (b) letting itbe closed. We will not try to express the idea of the agent letting thedoor be open or making it closed, if in fact the door is already in aclosed state. Thus the language we develop here is that of a special kindof action where the agent is alone in a situation, and can effect a changein that situation or allow the situation to remain as it is.

The more general accounts of action utilizing the 'bringing-about'notion, those of St. Anselm in Henry (1967), and those of (1977)and Walton (1979), all make a distinction between an agent making itfalse that p and not making it true that p. The account given here


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simply does not have the expressive capacity to straightforwardly makethe distinction. In other words, the present account is based on a strongassumption of asymmetry between active making false and passiveomission.

The language developed here is best applicable to the cases wherethe agent deliberates on whether or not to act so as to change an alreadygiven situation. For example, suppose I am deliberating on whether todig a hole in my backyard, and the present situation is that there is nohole there. On the present account, I can make it true that there is ahole or let it remain the case that there is no hole. I can't make it falsethat there is a hole there, or let it be true that there is a hole there. Ourlanguage is therefore one of action relative to a given situation.

2. EXTERNAL SEQUENCES OF ACTIONS

By the foregoing type of analysis, we can say that the externalact-sequence breaks down into a set of four propositions, each madetrue by Smith: Smith's finger was moved, the switch was flipped, thelight was turned on, the prowler was warned. Accordingly, each stage inthe sequence is a proposition, and accordingly we now re-ask ourquestion above as follows: what does it mean to say that eachproposition "has something to do with" each other proposition in thisset? Readers of Walton (1979) know that the answer is supplied byintroducing a relatedness relation that is reflexive and symmetrical,but not transitive: (p, q) means approximate spatio-temporal coin-cidence of p and q in the sense that p can affect q or q can affect p.Then po is indirectly related to pn+1 if, and only if, there are propositionsp1, p2, . . ., pn-1, pn such that po is related to p1, p1 is related to p2, . . . ,pn-1 is related to pn, and pn is related to pn-1. Propositions connected ina sequence by intervening related propositions are themselves in-directly related.

The specific context of deliberation about action we propose tomodel is that of an agent in a given situation confronted by a set ofpropositions. The situation is divided by the agent into a number ofspace-time zones or locations. Each atomic proposition takes on asubset of these sectors, assigned to it by the agent. The set of locationsof a proposition is called its zone or aegis. . If pi and pj share somelocations in common between their respective zones, we say that p i andpj are related, meaning that making one true can affect making the

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other true. Thus the structure of how complex propositions are madetrue or allowed to be false is mediated through the spatio-temporallocations assigned to these propositions by the agent of change in asituation.

We must be careful here to make a clear distinction between atomicpropositions and the complex propositions formed by connectives fromatomic propositions. Using capital letters, A, B, C, . . . , for complexpropositions, our problem is to know how to assign sets of locations tothese, based on our way of assigning sets of locations to atomicpropositions, p1, p2, ..., pn. We have to be careful to make thisdistinction clearly. For example, relatedness could be transitive overthe atomic propositions, but we will see that it can never be transitiveover complex propositions.

We must also be careful that what is meant by a spatio-temporal zonemay be complex in certain instances. For example, as Cresswell (1979)notes, 'it is raining in Wellington' does not mean rain is present in allpoints in Wellington but only in some acceptable subset. For the presenthowever, we overlook these subtleties of language.

We define a proposition as taking on precisely one of two values. Wei nterpret p = T as meaning that what some agent does makes p true.We interpret p = F as meaning that what some agent does lets p befalse. In constructing complex propositions, we need to take intoaccount both the "truth-values" and the question of how the pro-positions are related to each other.3 For any complex proposition "Aand B", we can say that A is made true and B is made true if, and onlyif, both A and B are made true. Similarly for negation, relatedness doesnot seem to matter, so we can define A in the classical way too. A ismade true if, and only if, it is not the case that A is allowed to be false.And A is allowed to be false if, and only if, A is not made to be true. Butit is when we come to define the conditional that relatedness makes adifference.

An external act-sequence is sometimes a series of conditionals. Forexample, we may presume it is being claimed that "If it is made true thatthe switch is flipped, then it is made true that the light is turned on" istrue for the sequence given above. Not that anyone need claim that it isl ogically necessary that if A (the switch is flipped) is made true then B(the light is turned on) is made true. But rather we do sometimespresume in certain act-situations that one claims as a matter of fact, orhindsight, that it is not the case both that A is made true and B is

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allowed to he false by the agent. As Segerberg (1982) points out,however, 'do' is ambiguous. Sometimes, 'the agent does something'i mplies that success was certain in what he did, other times not. IIere wemean to imply that when B is made true by making A true, then it is notthe case that A is made true and B is allowed to be false. However, insuch a claim more than this is presumed: we also in the case of anact-sequence require that A has something to do with B, i.e., A and Bare related. Hence we define A B (If A then B) nonclassically asfollows: A B iff and (A, B). We interpret A B asmeaning: whatever makes A true also makes B true. This definition wasin certain respects inspired by the conditional defined i n van Fraassen(1969, p. 485), but t here are important differences between vanFraassen's conditional and those analysed here.4

We now must decide how the complex conditionals are related.When, in general, is C related to A B? Consider "If the switch isflipped then the light is turned on." It seems reasonable to think thatthis conditional is related to "The bulb is working", and also to "Theswitch is working." Thus the best requirement is this: C is related toA B if, and only if, C is related to A or C is related to B. We can nowsee why relatedness on complex propositions can never be transitive.Since A is always related to A B and A B is always related to B, itwould follow by transitivity of that A is always related to B. By sucha proposal, every complex proposition would be related to every otherone.

Finally, we have to decide whether to define disjunction in such a wayas to require relatedness or not. We reason that A B is not made to betrue if A alone is made to be true and A has nothing to do with B. Thuswe define disjunction as follows. A v B is made to be true if, and only if,at least one of A or B is made to be true and A is related to B.

Now let us look to see which forms of inference are valid or not, forthe requirements set out above are enough to determine a class ofpropositional calculi. The tautologies for the requirements above turnout to he precisely those of System of Epstein (1979). A decisionprocedure in the form of modified truth-tables is available, requiringthat we take relatedness into account as well as truth-values.

First, modus ponens is valid: if A is made true, then if whatevermakes A true makes B true, then B is made true. For example, supposethat it is made true that the switch is flipped. Then if whatever makes ittrue that the switch is flipped makes it true that the light is on, then it is

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made true that the light is on. The following truth-table demonstratesvalidity.

The truth-tables are similar to classical logic except that we have totake into account all possible relatedness relations on the basic prop-ositions, as well as all possible combinations of truth-values.Readers of Epstein (1979) will know that the System that resultsfrom our requirements above is a subsystem of classical PC. Listedbelow are some tautologies that are shared with classical logic.

Some Tautologies of System of Relatedness Logic

Perhaps some of these theses are worth special comments. Let us takean example of (2), modus tollens. If the light is allowed to be off then ifwhatever makes the switch flipped makes the light go on, then theswitch is allowed to be not flipped. This inference is reasonable, forassume that the switch is not allowed to be not flipped. Then it mustfollow either that it's not made true that whatever makes the switchflipped makes the light go on or that its's not made true that the light isallowed to be off.

Contraposition, (8) also seems reasonable. Assume that whatever


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makes the switch flipped makes the light go on. Then whatever allowsthe light to be off also allows the switch to be not flipped.4a

Reflectionshould indicate that the remaining inferences are also correct forexternal act-sequences.

Perhaps even more interesting is that many inferences we would takenot to be applicable to conditionals in act-sequences, must fail on thepresent interpretation. Consider (1) from the set given below. Whatevermakes A true need not make it true that whatever makes B true makesA true. Reason: B need have nothing to do with what makes A true.Similarly with (8): whatever makes A true need not make it true that Aor B.

(L)(M)

true.

Some Tautologies of Classical PC That Fail in

Failure of transitivity might be taken to be an unfavourable charac-teristic for the present analysis of extrinsic act-sequences. For if the firstpair of conditionals below is made true, then surely the third is therebymade

If the switch is flipped, the light is turned on.If the light is turned on, the prowler is warned.If the switch is flipped, the prowler is warned.

Our reasoning here is that the third conditional need not be true in thesense that making it true that the switch is flipped is directly related tomaking it true that the prowler is warned. Still, the latter pair ofpropositions is indirectly related, and thus in that limited sense of ,(M) does follow (indirectly) from (K). But we cannot allow unrestrictedtransitivity, for the reason that remote consequences of actions may notbe regarded as part of the act-sequence at all, either directly orindirectly.

(K)


Let us assume for example that it is not the case both that the switchwas flipped and that the prowler's brother-in-law tripped over hislawn-mower and broke his leg two weeks later. Still, we need not saythat by making it true that the switch was flipped, Smith made it truethat the prowler's brother-in-law broke his leg. We might, but it woulddepend on how or whether these two situations were related.

Although transitivity does not generally obtain, we should notice thata close analogy to transitivity may be added to the set of tautologies for

, namely the following schema:

[(A B) (B C) A] C.

We can show this as follows. In order for (21) not to hold, C must havethe value F and each of the three conjuncts of the antecedent musthave the value T. So A must have the value T, and therefore forA B to take the value T, B must take the value T as well. But thesevalues, namely B = T and C = F, make the remaining conjunct,B C, take the value F.

So because of these truth-values, (21) is a relatedness tautology aswell as a tautology of classical logic. So if we have (K) and (L) as above,with 'The switch is flipped' as a premiss, then the conclusion 'Theprowler is warned' follows deductively. However, the implication hereis reasonable as long as we realize that the last pair of statements quotedabove do not have to be directly related to each other. Perhaps weshould add parenthetically that (21) would not be a relatedness tau-tology if we defined conjunction so as to require relatedness, as shownpossible by Epstein

(1979). But that is not a view of conjunction wewant or need to advocate for act-sequences.5

Several of the last list of schemata, like (1) and (2), are ones that haveoften been rejected as "paradoxical", but (10), even though highlyparadoxical even in classical logic in its usual truth-theoretic inter-pretation, does not appear to have been considered as troublesome, bycomparison to (1), (2), or (3).

Consider an example. If I flip both switch A and switch B then thelight will go on. Therefore, at least one of the following is the case: if Iflip A the light will go on, or if I flip B the light will go on. Thisinference would fail if both switches together were required to make thelight go on. So it seems the schema should not be generally true fori nternal act-sequences. Yet it is easily seen that it is a classical

(21)


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tautology, i.e., that is a tautology of classical PC.

An example will illustrate how deeply problematic this is for deduc-tive implication. Consider any deductively valid argument where bothpremisses are required.

Pierre is taller than Quetzil.Quetzil is taller than Rudolf.Therefore, Pierre is taller than Rudolf.

That is, if either premiss is deleted, the remaining one does not by itselfdeductively imply the conclusion. The problem is: if you accept theschema above as a principle of classical inference, then you have toaccept that if the above argument is valid then at least one of thepremisses must by itself deductively imply the conclusion. That is,given the validity of the above argument, it follows that at least one ofthese arguments must be valid.

Pierre is taller than Quetzil.

Quetzil is taller than Rudolf.Therefore Pierre is taller than

Therefore Pierre is taller thanRudolf.

Rudolf.

But this is indeed sophismatical. For surely the first argument isdeductively valid, but neither of the remaining pair is by itself adeductively valid argument. What is wrong? A good question, forstudents of propositional calculi, but suffice it to say here that anytheory of act-sequences based on the classical logic of propositions hasto confront the question of how to deal with this sort of inference.

(1977) bases his theory of actions on classical propositionalcalculus. He introduces a modal operator 'it is necessary for somethingthe agent does that p' where p is a proposition that obeys the laws ofclassical PC. In line with the present discussion of conditionals, let usturn around 's basic expression to an equivalent 'something theagent does is

sufficient for p', and confront the following problem.

Suppose an agent constructs a machine in such a fashion that if bothswitches A and B are flipped then light C will go on. But suppose heconstructs the machine so that both switches are required to be on forthe light to be on. That is, if either is off, the light will not be on. Using

's language, we say that something he does, namely constructingthe machine, is sufficient for the following to obtain: if A and B areboth on then C is on. By the logic of 's operator we must

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deductively infer that the following also obtains: it is either the case thatif

A is on C is on, or it is the case that if B is on C is on.6 But does itreally follow?

Our very description of the agent's construction of his machineassures us that the premiss is true but the conclusion is false. For it is notthe case that if A is on C is on, and it is not the case that if B is on C ison. So it would seem that any analysis of 'if ... then' based on classicallogic would have to deal with this dubious implication.

3. INTERNAL SEQUENCES OF ACTIONS

Some stages in an act-sequence are related to other stages as we shallnow say, internally. Consider this familiar sequence: Buttering thetoast, Smith's buttering the toast, Smith's buttering the toast slowly.Smith's buttering the toast slowly and deliberately, Smith's butteringthe toast slowly and deliberately with a knife, Smith's buttering the toastslowly and deliberately with a knife in the bathroom, Smith's butteringthe toast slowly and deliberately with a knife in the bathroom at mid-night. This sequence falls naturally into the order in which it is givenabove. The rationale of the sequence would seem to be that as each stepi n the description of the complex of events is taken, more informationabout the development of what happened is given. In each subsequentcase, the development of the sequence as described is more specific asan unfolding situation. That is, there is a sense in which each stage ofthe unfolding sequence is included in each other stage that comes afterit.

We start with the idea that each of the descriptions above containssome items of information telling us how something is made true. Thefirst description of the event contains the information that it was abuttering and that the toast is what was buttered. The second descrip-tion adds to the information in the first by telling us that the aforesaidtoast-buttering is Smith's. It brings forward the previous information,but at the same time adds a new element. The transformation from thefirst to the second step is one of increased specificity of informationabout how something was made true. The information contained in thefirst step is included in the information contained in the second step.The converse does not obtain. That is, information content inclusion ininternal event sequences is not a symmetrical relation. Clearly,


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however, it is a reflexive relation, as we must say that the informationcontent in any step of the sequence is contained in that very step itself.

The third description brings in yet another distinct item of in-formation, namely that the event described in step two is one that tookplace slowly. Thus the information content of step three includes that ofstep two. We have already established that two includes one. Now wecan see that informational content inclusion in such a sequence istransitive, and we may conclude that the information in step three alsoincludes that given in step one.

Similar reasoning applies to all seven steps in this sequence. Eachsubsequent step includes all of its predecessors. The seventh step istherefore related by informational inclusion to every step in thesequence. Thus the relation of informational inclusion effects a linearordering of the sequence.

As a proposition that contains information about how an eventhappened, 'Smith buttered the toast slowly' is more specific, i.e.,contains more information about how it was made true that the toastwas buttered than 'Smith buttered the toast'. Slowly adds a newelement, and tells you more about what happened and how it happened.In this sense of informational containment, we say that 'buttered slowly'contain '' buttered'.

Smith buttered the toast slowly

Smith buttered the toast

We look at it this way insofar as we regard these two propositions asexpressing different degrees of information about what was made to betrue.

But there is an ambiguity. If we look at the different ways these twopropositions could come to be true, it seems that the information that itis possible that Smith buttered the toast slowly is contained in theinformation that it is possible that Smith buttered the toast. It's not thecase that every way in which he could butter the toast is a way in whichhe buttered it slowly, e.g., he could butter it quickly. But the converse is

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true: every way in which he butters the toast slowly is a way in which hebutters the toast. In this sense 'Smith buttered the toast' contains moreinformation about the possible ways things could have happened.

With a sequence like the one we began with however, it seems thatour concern is more with what information is given about what wasmade true and how it was made true, than with the different possibilitiesof how it could have come to be true. However, we will return to thisalternative notion of informational containment of possible ways thingscould have happened.

The general outline of our present approach is this. Given that wecan start with such a set of bits of information about a sequence ofactions, we can define the information content of any particularproposition made true at some stage as a subset of this large set webegan with.7 Let us call the set of bits of information about how eachproposition was made true I. Then we are saying that i(p) for anyproposition made true in the sequence is some subset of I. Propercontainment is not always required. Thus i(p) can be I itself in somecases. Generally, if a sequence is made up of propositions P0, P1, . . . ,Pn , we will say that the informational content of that sequence is theunion of the information content of all its components

The requirements we have now set down are sufficient to assure us thatthe correct approach to reasoning about internal action sequences canbe modelled by a dependence logic of Epstein (1982). FollowingEpstein, we can introduce a binary relation of information-inclusion onthe complex of propositions, called a dependence relation, where there issome I and i as above such that B is included in A if and only if

Epstein's definition of a model takes into account both atruth valuation and a dependence relation. Consequently the way wewant to define conditionals in an act-sequence as above is possible in adependence logic. Some examples of tautologies are given below.

The kind of conditional involved, 'if A then B', requires bothto be made true. The first requirement is

satisfied by the pair of propositions; A = Bob swallowed a sword,B = 2 + 2 = 4. Presuming B is true then is satisfied simplybecause B is true, B is satisfied, and therefore is alsosatisfied. The second requirement is satisfied by the pair of propositions;

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A = Bob swallowed an imaginary sword, B = Bob swallowed a sword.All the information in B is contained in the information of A. Henceneither requirement alone is sufficient to determine the truth of 'If Athen B' in the sense required for internal act-sequence inferences.

To determine that 'Bob swallowed a steel sword' forms the ante-cedent of a true conditional where 'Bob swallowed a sword' is theconsequent, we must require both the right truth-values and the rightinformation-inclusion relationship of these propositions. Neither byitself is adequate.

Neither informational containment nor truth-values of individualpropositions is by itself sufficient to account for the kind of conditionalinvolved in an internal act-sequence. Just using individual truth-values,we can truly say that if 'The toast is buttered' is allowed to be false thenthe following conditional must be true: whatever makes it true that thetoast is buttered makes it true that Hannibal was defeated at theTrasimene Lake by the Romans. Just using information containment,we have to say that the information concerning how 'The toast isbuttered' is made true is contained in the information on how 'The toastis buttered at midnight with a knife'. Yet clearly it is incorrect toventure that whatever makes the second proposition true makes thefirst proposition true. Thus the kind of conditional involved in internalact-sequences requires both and to be made true.

Here are some examples of tautologies that result from this accountof conditionals.

Some Tautologies of Dependence Logic

To contrast the logic of internal versus external sequence of actions,note that (17) above holds in the dependence logic but fails in arelatedness logic. 'If the toast was buttered slowly at midnight, then thetoast was buttered slowly and the toast was buttered at midnight'


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implies 'If the toast was buttered slowly at midnight then the toast wasbuttered slowly, and if the toast was buttered slowly at midnight thenthe toast was buttered at midnight.'

The general reasoning is as follows. Look at the consequent of (17).There are only four possible ways it could be allowed to be false: (i) if Ais made true and B is allowed to be false, (ii) if B is not contained in A,(iii) if A is made true and C is allowed to be false, or (iv) if C is notcontained in A. But each of these interpretations will make A( B C), the antecedent of (17), allowed to be false. Clearly either (i) or(iii) would make A (B C) false, just in virtue of the truth-values.But either of (ii) or (iv) would have the same effect, as we can see byconsidering (ii). If B is not contained in A, then clearly B C will notbe contained in A either. In short, every possible way of allowing theconsequent of (17) to be false bars the possibility of making theantecedent true. Thus there is no consistent way to assign truth-valuesand information content assignments so that A (B C) is made trueand (A B) ( A C) is allowed to be false. So (17) is a tautology forcontent inclusion of act-sequences.

Yet for relatedness, (17) fails to be generally true. To disprove it,suppose A is related to B but not to C. Then A C is allowed to befalse, and hence ( A B ) (A C) is allowed to be false. But A

(B C) could still be made true. Assume B and C both are allowed tobe true. Then A ( B C) is allowed to be true, because A is related toB C and B and C are allowed to be true. Then A (B C) is made

to be true but (A B) ( A C) is not. Thus (17) fails for relatednessof events. For example, let us assume that whatever makes it true thatmy finger is moved makes it true that the switch is flipped and the lightis on. It need not follow that my moving my finger by itself makes it truethat the light is on. It need not follow that just because I move my fingerthat by some sort of wizardry the light must go on.

Below are some formulas that are not tautologies in dependencelogics.

Some Tautologies of Classical PC That Fail in Dependence Logics


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Take (2) as an example. Whatever makes it true that the toast isbuttered need not also make it true that the toast is buttered or the toastis eaten. Or consider (12). Whatever makes it true that the toast isbuttered need not make it true that if the toast is eaten it is buttered.

As an illustration to show how Venn diagrams can be used along withtruth-tables as a method of testing tautologies in dependence logic, letus look at the case of exportation. First consider

This schema is a classical tautology, but is it also a dependencetautology? For the antecedent to be true, it is required thati(A), i.e.,

For the consequent to be true, it is required that Thisrequirement has to be met, as the diagram shows that all information inC, but outside A B, is nullified. Hence the above schema is adependence tautology. What about the other way?

The antecedent requires But as the diagram belowi ndicates, that (toes not mean that

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DOUGLAS N. WALTON

There could be some information in B C that might not necessarilybe included in the information in A. Thus by looking at all possiblecombinations of truth-values and all possible combinations of in-formation overlap for the given set of atomic propositions, we canalways determine whether a given wff is or is not a dependencetautology. The second schema fails to be a dependence tautology eventhough it is a tautology in classical propositional calculus.

4. ALTERNATIVE LINES OF HISTORICAL VERIFICATION

Another thing we could mean by A B is that every way in which A

could be known to have come to be true is also a way that B could beknown to have come to be true. For example, every way in which 'Smithbuttered the toast slowly' could have known to have come to be true is away in which 'Smith buttered the toast' could have known to become tobe true. But the converse is not true. For some ways Smith could havebeen known to have come to have buttered the toast, e.g., quickly. arenot ways he could have been known to have come to butter it slowly. Inthis sense, 'Smith buttered the toast' contains more information abouthow things could have happened.

Smith buttered the toast quickly (C).

Smith buttered the toast (B).

Smith buttered the toast slowly (A).

In this sense, the more general proposition contains more informationabout the different historically possible ways it could have becomeknown to be true.

In this example there are four ways A could be known to have cometo be true.


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All the different ways that A could have become known to be true are

also ways that B could have become known to be true.According to this way of ordering the sequence of actions, the

relation of information-inclusion is just the converse of the relation wehave been studying in internal sequences. Hence the appropriate logicfor this structure of internal sequences is a dual dependence logic ofEpstein (1982). In keeping with this approach to actions, we might wantto study internal sequences of actions by looking at the different worldlines that represent the historical ways that the proposition could havebeen known to have come to be true. But note that this approach, thedual of our previous approach to internal act-sequences, tends to makethe sequence more pluralistic. For if we are talking of Smith's butteringthe toast slowly as opposed to quickly, both as being instances of but-tering, it seems more plausible that we are talking of different historicalpossibilities of actions. Thus here the notion of an act-sequence yieldsinformation about different historical possibilities i n a line of develop-ment. Really what we are investigating here is the epistemology ofhistorical verification concerning how we come to know that a givenaction occurred. We leave the investigation of these conditionals foranother occasion. It is enough to notice that this way of ordering actionsis quite different from either of the previous two ways.

Some of the most fascinating open questions concern negation inact-sequences. In dual dependence logic, i(A) i(B). So (8), (9), and (l6) i n the list of tautologies for depen-dence logic will fail in dual dependence logic. Consider (8):

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Parenthetically, we might note that this fact has important con-sequences for the Raven Paradox.8 Every way it could become knownto have come to be true that if Bob is a raven then Bob is black, neednot be a way it could become to be known to be true that if Bob is notblack then Bob is not a raven.Epstein (1982) has worked out a number of l ogics in which theconditions jointly obtain. Thesecond condition means that, unlike all the systems we have consideredabove, Moreover, in some modal logics, both conditionsabove are met. These developments open up new alternatives instudying the logic of act-sequences, and suggest (i) letting a propositionbe false may, in some important sense, be less specific in informationcontent than making the same proposition true, (ii) negation may be apromising avenue of new developments, and (iii) extensions to modall ogic may bring out richer theories of the distinction between actionsand omissions like the modal theory of Talja (1983). For the present, wenow turn to a specific problem.

Cohen's problem of (1971, p. 63) can be stated as follows. Considerthe following pair of conditionals.

(C 1)

If the government falls, there will be rioting in the streets.( C2)

If it is the case both that if the government falls there will berioting in the streets, and also that the government will notfall, then the shopkeepers will be glad.

Let us say that (C1) and (C2) respectively have the forms 'A B' and Now here is the problem. Assume A is false.Then by the truth-functional readings ofare both true. Hence (C2) as a whole must be true. What is problematic,according to Cohen, is that the truth of the consequent of (C2), thestatement that the shopkeepers will be glad, is assumed to be dependenton only one condition, the fate of the government.9 Whereas really inasserting (C2), we are saying that it is dependent on two mutuallyindependent conditions. What is wrong, according to Cohen (1971, p.63), indicates that 'if ... then' cannot be truth-functional, even despite

DOUGLAS N. WALTON

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the attempts of conversationalist theorists to patch up the truth-functionalist analysis.

Cohen's problem does not arise on a relatedness analysis of (C1) and( C2), for letting A be false is not enough by itself to guarantee that(A B) A is allowed to be false. Such an assertion is only made if Ais unrelated to B. Thus in the relatedness analysis, making C true isdependent on the two mutually independent conditions A B and A.

Moreover, other problems do not arise as well. Even if A B and

A were both allowed to be false, it still does not follow automaticallythat (C2) is true and that 'The shopkeepers will be glad' has to be madetrue regardless of its connections to

A B or A. In order for (C2) t obe made true, the consequent (C) does have to be related to at least oneof the propositions in the antecedent.

By requiring relatedness of propositions made true or allowed to befalse, relatedness logic provides an analysis of action inferences that ismore comprehensive than an exclusively truth-functional analysis. Inrealistic argumentation of the ad hominem sort, the disputation oftenturns on relatedness or dependence relations that are quite complex.The

links of relatedness in a sequence of actions is often itself a topic of

dispute. Suppose Jones accuses Smith of selling weapons to a repressiveregime. Then Smith replies, "Why, you're inconsistent and hypocritical.The university you are employed by has investments in companies thatmanufacture the very weapons used by this same regime." The al-legation is that there is a sequence of links that could be filled in so thatwhat Jones allows is a proposition related to some outcome

equivalent

to what Jones deplores. The allegation presumes an analysis that, iffilled in, could support a demonstration of an act-theoretic incon-sistency on the part of Jones. Whether the criticism of ad hominem hypocrisy is tenable or not must be fought out, we propose, by a fillingi n of the propositional logic once Smith and Jones agree on theirpremisses. However, we must l eave the full analysis of such complexdisputations for another study.

NOTES

* I would like to acknowledge support through research grants from the Social Sciencesand Humanities Research Council of Canada and the University of Winnipeg, and thankDick Epstein for criticisms of a previous draft of this paper.


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1 See also Howard L. Dazeley and Wolfgang L. Gombocz: 1 979, 'Interpreting Anselm asLogician'. Synthese 40, 71-96.

2 For further remarks on action propositions, see the author's critical notice of(1977) in Synthese 43, 1980, 421-431.3 See the definition of a model for relatedness logic given by Epstein (1979, p. 148). Thedevelopment in section 2 could be re-worked for a non-symmetrical , but we shouldnote that contraposition will fail and A B will be defined as(R(A, B) A R(B, A). Note that capital R is syntactic and script is semantic.4 Some comparisons of relatedness logic and relevance logic are discussed in an article bythe author, 'Philosophical Basis of Relatedness Logic', Philosophical Studies 36, 1979,115-136.4a Symmetry of could be dropped, except that it's needed to derive contraposition.

The discussion in section 2 could easily be reworked for nonsymmetrical , butcontraposition, (8), will fail.

5 Failure of transitivity in is closely connected to failure of a deduction theorem: in ,just because we can prove B from A, it does not follow that A B must obtain. In , ifwe can prove B from A and we can prove C from B, t hen we can always prove C from A.Yet transitivity for does not obtain.6 However, also introduces other operators in his language that could possibly alsobe equated with our notion of making a proposition true. For further discussion see thereference of note 2 above.7 There will be many cases where it is difficult to straightforwardly assign an informationset to a given proposition taken by itself, but given a set of propositions reconstructedfrom an act-sequence, we can argue relative information more easily.8 For a statement of the Raven Paradox, see Carl G. Hempel: 1965, 'Studies in the Logicof Confirmation', in C. G. Hempel, Aspects of Scientific Explanation, The Free Press, NewYork, pp. 3-51. Reprinted from Mind 54, 1945, 1-26 and 97-121.9 You can see why by the following reasoning. If A is false then must betrue. If A is true then must be false. Thus as Cohen (1971, p. 63) pointsout, evidence for the truth of (C2) is sufficient if it relates the fate of the government tothe feelings of the shopkeepers "without having any bearing whatever on the causes andeffects of rioting in the streets."

L. J. Cohen: 1971, 'Some Remarks on Grice's Views about the Logical Particles ofNatural Language', in Y. Bar-Hillel (ed.), Pragmatics of Natural Languages, Rcidel,Dordrecht, pp. 50-68.

M. J. Cresswell: 1979, 'Adverbs of Space and Time'. in F. Guenther and S. J. Schmidt(eds.), Formal Semantics and Pragmatics for Natural Languages, Reidel, Dordrecht, pp.171-199.

D. Davidson: 1967, 'The Logical Form of Action Sentences', in N. Rescher (ed.), TheLogic of Decision and Action , University of Pittsburgh Press. Pittsburgh, pp. 81-95.

R. L. Epstein: 1979, 'Relatedness and Implication', Philosophical Studies 36, 137-173.R. L. Epstein: 1982, ' Relatedness and Dependence in Prepositional Logics'. Research

Report of the Iowa State University Logic Group.

DOUGLAS N. WALTON

REFERENCES

A. I. Goldman: 1970, A Theory of Human Action, Prentice-Ilall, Englewood Cliffs, NJ.D. P. Henry: 1967, The Logic of St. Anselm, Clarendon Press, Oxford.I. : : 1977, Action Theory and Social Sciences: Some Formal Models, Reidel,

Dordrecht.B. C. van Fraassen: 1969, 'Facts and Tautological Entailments', Journal of Philosophy 66,

477-487.K. Segerherg: 1981, 'Action-Games', Acta Philosophica Fennica 32, 220-231.J. Talja: 1983, 'A Logic of Omissions'. Reports from the Department of Theoretical

Philosophy, University of Turku, No. 11.D. N. Walton: 1979, 'Relatedness in Intensional Action Chains', Philosophical Studies 36,

175-223.D. N. Walton: 1980, 'On the Logical Form of Some Common-place Action Expressions',

Grazer Philosophische Studien 10, 141-148.

Department of PhilosophyUniversity of WinnipegWinnipeg R3B 2E9Canada


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